Generalized Permutations and The
Multinomial Theorem
1 / 19
Overview
The Binomial Theorem
Generalized Permutations
The Multinomial Theorem
Circular and Ring Permutations
2 / 19
Outline
The Binomial Theorem
Generalized Permutations
The Multinomial Theorem
Circular and Ring Permutations
3 / 19
The Binomial Theorem
Theorem
(x + y )n =
n
X
C (n, r ) x n−r y r
r =0
4 / 19
Binary Sequences
Count the number of binary sequences of length n in two
different ways.
5 / 19
Outline
The Binomial Theorem
Generalized Permutations
The Multinomial Theorem
Circular and Ring Permutations
6 / 19
Generalized Permutations
Definition
Let X be a set of n not necessarily distinct objects belonging
to k different nonempty groups such that
1. all the objects in a group are identical
2. an object in a group is not identical to an object in
another group .
A generalized permutation of X is an arrangement in a row of
the n objects of X .
Anagrams are generalized permutations. A famous
contemporary example:
IAMLORDVOLDEMORT
TOMMARVOLORIDDLE
7 / 19
The Number of Anagrams
Theorem
If the set X of n objects consists of k different nonempty
groups such that group i has ni identical objects for 1 ≤ i ≤ k,
then the number of generalized permutations of X is
n!
.
(n1 !)(n2 !) · · · (nk !)
[anagram tool]
Example
Determine the number of generalized permutations of the 5
letters that appear in the word LEMMA.
8 / 19
The Number of Anagrams
Theorem
If the set X of n objects consists of k different nonempty
groups such that group i has ni identical objects for 1 ≤ i ≤ k,
then the number of generalized permutations of X is
n!
.
(n1 !)(n2 !) · · · (nk !)
[anagram tool]
Example
Determine the number of generalized permutations of the 6
letters that appear in the word TSETSE.
9 / 19
Some Identities
Definition
P(n; n1 , n2 , . . . , nk ) :=
P(n,n1 +···+nk )
(n1 !)(n2 !)···(nk !)
Proposition
We have the following combinatorial identities:
1. P(n; r ) = P(n; n − r ) = P(n; r , n − r )
2. P(n; r ) =
P(n,r )
r!
.
10 / 19
The Allocation Interpretation of Generalized
Permutations
Theorem
If there are ni identical objects in group i for 1 ≤ i ≤ k and if
r = n1 + · · · + nk is the total number of the objects in these k
groups, then these r objects can be placed in n distinct
locations so that each location receives at most one object in
P(n; n1 , n2 , . . . , nk ) ways.
In particular, if each group has exactly one object, then this
number of allocations is P(n, r ).
11 / 19
Outline
The Binomial Theorem
Generalized Permutations
The Multinomial Theorem
Circular and Ring Permutations
12 / 19
The Multinomial Theorem
Theorem
In a typical term of the expansion of
(x1 + x2 + · · · + xk )n
the variable xi appears ni times (where n1 + n2 + · · · + nk = n)
and the coefficient of this typical term is
P(n; n1 , n2 , . . . , nk ) =
n!
.
(n1 !)(n2 !) · · · (nk !)
13 / 19
Outline
The Binomial Theorem
Generalized Permutations
The Multinomial Theorem
Circular and Ring Permutations
14 / 19
Circular Permutations
Circular permutations are a variant of the r -permutations of a
set X of n distinct elements we have been considering.
Suppose that we now assume that two permutations are the
same provided that one can be obtained from the other by
cycling.
For example, the 3-permutations of the set X = {A, B, C }
given by ABC, CAB, and BCA are the same when considered
as circular permutations.
15 / 19
Circular Permutations
Circular permutations are a variant of the r -permutations of a
set X of n distinct elements we have been considering.
Suppose that we now assume that two permutations are the
same provided that one can be obtained from the other by
cycling.
For example, the 3-permutations of the set X = {A, B, C }
given by ABC, CAB, and BCA are the same when considered
as circular permutations.
Proposition
The number of circular permutations of a set of n elements is
P(n, n)
= (n − 1)!.
n
16 / 19
Ring Permutations
Supposing that two permutations are the same provided that
one can be obtained from the other by cycling or by mirror
reversal, we obtain the notion of a ring permutation.
For example, the 3-permutations of the set X = {A, B, C }
given by ABC, CAB, BCA, CBA, BAC, and ACB are the same
when considered as ring permutations.
17 / 19
Ring Permutations
Supposing that two permutations are the same provided that
one can be obtained from the other by cycling or by mirror
reversal, we obtain the notion of a ring permutation.
For example, the 3-permutations of the set X = {A, B, C }
given by ABC, CAB, BCA, CBA, BAC, and ACB are the same
when considered as ring permutations.
Proposition
The number of ring permutations of a set of n elements is
1 P(n, n)
(n − 1)!
·
=
.
2
n
2
18 / 19
Acknowledgements
Statements of results follow the notation and wording of
Balakrishnan’s Introductory Discrete Mathematics.
Some examples follow Rosen’s Discrete Mathematics and Its
Applications.
19 / 19